Theoretical and Applied Genetics 9 by Springer-Verlag 1976

48, 67-73

(1976)

Efficiency of Selection in Layer-type Chickens by Using Supplementary Information on Feed Consumption I. Selection Index Theory* C.R.

Arboleda,

D.L.

Dept.

of Animal

Science,

Harris

and A.W.

Iowa

Nordskog

State University

of Science

and Technology,

Ames,

Iowa

(USA)

Summary. Selection indexes to maximize net income for egg laying chickens were constructed with information on egg mass output, body weight and individual feed records. Relative selection efficiencies were then compared with different kinds of information in the index. If the genetic variation in feed consumption is completely determined by egg mass output (M) and body weight ( W ), using reliable estimates of genetic correlations or pehnotypic regressions of these traits with feed consumption in the index is equally efficient to an index with individual feed records. If real genetic differences in feed efficiency exist which are independent of egg mass and body weight, (h~2), then there is greater justification in using individual feed consumption records. For example, if h~ = 0.2, h~ = 0.6, hM2 = 0.15 and r G (genetic correlation) = 0.2, the use of individual feed records is expected to improve efficienWM

cy of the selection for net income by 9p. On the other hand, if the genetic correlations of feed consumption on body weight and on egg mass are substituted in the index for records on individual feed consumption, only slightly less selection efficiency would result.

Introduction Even

proved

though

feed represents

cost of egg production, variation indeed.

This

Rather,

research evaluation

would

devoted

because

mation

weight.

the measurement records

net performance,

in large num-

or net income

has three basic options (I) records

traits and feed consumption, related traits supplemented on feed consumption,

and

related traits. The basic ficiency of a selection net income

when

in his choice

the of se-

F (Option

question

only on income

is what is the ef-

directed

feed consumption

towards

in-

is known

in terms

index theo-

of selection

we shall specify the conditions index for net income

un-

can be im-

W,

informa-

I). Y2

M

with estimates

of W

and

M

with

W

and

M

W

and

M.

Y3

Y2

and

are variations total income,

and

M.

of each

I. Thus,

W, M, F)

M

using inforand feed coninformation

information

partial regressions

Y3

W

is designed

(IF),

supplements

to maximize

only on

YI

in ac-

of genetic correlations

supplements

with phenotypic

is designed

is Y(IF:

F.

developed

egg mass,

and

of Table

(3) records

feed costs

W

umn

conveniently

a selection

sumption, on

indexes

listed above.

over

weight,

(2) records

can be

der which

on body

description

it isn't. The problem

ry. In particular,

income

mation

relative to that when handled

with the options

hand

on income-

with supplementary

I lists 4 selection

on both income-related

with indirect information

program

Table

to maximize

mostly

much

tion on feed consumption.

cordance

few

be expensive.

lection criteria:

creasing

been very

of traits related to income

feed consumption

To improve breeder

have

viability and body

is understandable

of the

studies on individual

has been

as egg production,

of individual bers

genetic

in feed consumption

to the genetic such

about two-thirds

and by how

on

of F on

of Option

2. Y4

I, using infor-

To aid the reader,

a short-

index is given in the last col-

the shorthand

where

description

the trait maximized

left of the colon and the input information

of YI is to the

is indicated

on the right. Selection

Index Theory

The maximization

of genetic

feed cost is readily derived selection Hazel

index as developed

gains in net income in terms by Smith

over

of a standard (1939)

and

(1943).

Valid application

of selection

index theory

requires

that : * Journal Paper No. J-8192 of the Iowa Agriculture and Home Economics Experiment Station, Project 1711.

(a) the phenotypic as x i = gi + el' where

value of trait x i is expressible gi is the additive genetic

corn-

68

C.R.

Arboleda,

D.L.

Harris

Table I. Selection feed consumption

Selection index

portent and genetic

options

using

com-

or breeding value, H, of an individual is det fined as H : ~ aigi, where a. is the economic val1 i:1 ue of a unit of the i th trait

the

x's i function

is linear.

on any linear

function

of

selection is based on a linear t of xi, say, Y : ~ bixi, the expected imi:1 provement in H is AH : BHyAY , where Ay is the se-

Y. This may

~H(Y)

and

BHy

is the regression

of H

F

and of F

2

rHY~H(AY/~y)

:

on

M, M, M, M)

x3,

x I and body

of a hen

measured

population

and egg

of body

egg mass.

weight

and

The

the amount and

It is obvious no additive

from

netic parameters

B1

~2

the respective for main-

is a feed constant

quantity

to

u is a residual

of feed either

the model

component

wasted

re-

or used

given

in (1) that with

in the residual,

of feed consumption

weight

and

from

metabolism.

genetic

only of body

respectively,

~ 1 is a feed constant

produce

for digestion

of feed con-

mass,

as deviations

tenance

presenting

M

x 2 are the records

weight

means~

Chickens

F) r) b)

W and between F and on W and F on M

sumption,

when

also be written,

Cov(H,Y)hY/a

:

F rG bp

When

lection differential on

M, M, M, M

where

ponent,

of H

information

Y(IF:W, Y(IF:W, Y(IF:W, Y(I:W,

~~

(c) the regression

in Layer-type

W, W, W, W,

and non-additive

genetic

supplementary

of Selection

Descriptive notation of index

income over feed cost total income body weight egg mass feed consumption r -- genetic correlations between b = phenotypic partial regressions

(b) for trait i : 1,... ,t, the additive

Efficiency

Information required for the index

IF IF IF I

e i is the environmental

component

index

Nordskog:

Variable maximized

YI Y2 Y3 Y4 IF = I : W = M = F = G : : P

and A.W.

and egg mass

the ge-

are functions

and are estimable

~2 are known.

A p p l i c a t i o n to I m p r o v e I n c o m e O v e r F e e d C o s t The n e t b r e e d i n g v a l u e of i n c o m e o v e r f e e d c o s t , H,

If Y

is normally

without

error

in H,

then

intensity

standardized when

if the b's are

and if the top p proportion

by truncation, selection

distributed,

AY/~Y where

normal

= z/p

selection

= i is defined

is on

The

i s d e f i n e d a s a l i n e a r c o m b i n a t i o n of the b r e e d i n g

is selected

z is the ordinate

distribution.

known

as the

gain

of a model

statistical to represent

consumption (g3)' H = alg I + a2g 2 - a3g 3 where

model

is the simplest

feed consumption.

choice

a. is the economic i

If the breeding

(1)

of a unit of the ith

value

only of body

of feed consumption

weight

and egg

mass,

[g 3 = ~ig I + ~2g2 ] then the net breeding come

H~ Henceforth el will be called the environmental component but it will be understood that it contains also any non-additive genetic variance.

value

trait.

Let,

function

x 3 = ~1Xl + ~2x2 + u

(2)

=

= i Cov(H,Y)/~y. A linear

egg m a s s (g2) a n d feed

of the

expected

Y, is then AH(Y)

v a l u e s of body weight ( g l ) ,

over

feed cost equation

= alg I + a2g 2 - a3(~lg = (a I - a3~l)g

value

can be written,

I + ~292 )

I + (a 2 - a3~2)g

2

is a linear i.e., of in-

C.R. Arboleda,

D.L. Harris and A.W.

Finally, the net breeding terms

Nordskog:

Efficiency of Selection in Layer-type

value can be defined in

of total income,

69

i.e., g3 : ~Igl + ~2g2" This ignores any residual genetic variation for feed consumption. Relative Efficiencies

H~," : alg I + a2g 2. From

Chickens

If maximum

these breeding values we define four se-

improvement

in H is the principal ob-

jective in a breeding operation,

lection indexes :

gain, AH(YK), Index

Breeding

Y1 = b l l X l + b12X2 + b13X3'

value

in H from

then the expected

using selection index

is

Cov(H, Yk ) (Yks

H AH(Yk) :

Y2 : b21X1 + b22X2

'

H

Y3 : b31Xl + b32X2

'

H~

Y4 : b41Xl + b42X2

'

H~

YK'

2

iCov(H,Yk ) -~Yk ) :

cyk

~Yk

(3 )

where

Yk

: mean of the individuals s e l e c t e d on index Yk S

For each index the b's are chosen correlation mized.

between

The normal

the respective

such that the

bYk = population m e a n for index Yk

Y and H is maxi-

i = ( ~ k s - ~ Y k ) / a Y k is the s t a n d a r d i z e d s e l e c t i o n

equations for each of the indexes

are,

differential.

Index

YI:

Y2:

In m a t r i x notation, let

ipl112 pl i b111 11g12gllIalI P21

P22

P23

Ib12

P31

P32

P33

|b13

I~11 p,2 I P~I

P22

:

]ll, I~

g21

g22

g23

a2

I g31

g32

g33

-a3

: ~11 012 ~13 g21

g22

llall -a3

g23

H

:

a'g

YK : bkXk where

a and g are the column

vectors of economic

weights and of breeding

values of traits in H, respec-

tively ; b k is the column

vektor of coefficients corres-

ponding to the traits in the column ection index

Y3:

[P11 P12 l,b31] P21

P22

[b32

gll

g12

(~IglI + ~2g12 )

g21

g22

(91g12 + ~2g22 )

I

pose of the vector

iiall

(or matrix)

defined. The normal rYk H are (4)

so that

-a 3 bk= PklGk a

P22 I

b42

g21

g22

we collect no information

consumption

where

a2

we require full information

lize information

on W,

M

and

on F but we uti-

on the genetic correlations

of feed

is the i n v e r s e of the v a r i a n c e - e o v a r i a n c e

m a t r i x P k ' of the t r a i t s in Yk; Gk is the genetic v a r i a n e e - e o v a r i a n c e m a t r i x g e n e r a t e d by the e l e m e n t s in g and x k. F r o m equation 4, CoV(Yk,H) = b~Gka = b~Pkb K

on the other traits, i.e., g12 and g13'

in deriving the index. In Y3 completely

x k in sel-

Pkbk : G k a

,P1 Pi21[b41] [011 21[al ] F. In Y2

vector

(') indicates the trans-

equations obtained by maximizing

a2

Y4:[P21 In YI

Yk' The prime

determined

we assume

that g3 is

by body weight and egg mass,

2 : eyk

(5)

70

C.R. Arboleda,

From

equations

D.L. Harris and A.W.

Nordskog:

3 and 5, the expected gains in H

Coy ( Yk' H I )

G3

: i CY 1

AH(Y 2) = i

(6)

o211 2 g31 g21

~YI

Cov(Y 2, H 1 ) C~Y2

: i

(7)

~Y2

Because the coefficients of indexes Y3 and Y4 are obtained for alternative definitions of H(H* and H**), the identity given by (5) does not hold and the expected gain in H from these indexes can be expressed

only

g22

g23

i. e., there is no residual genetic component consumption, and since P3

as

AH(Y3) =

for feed

:2=IP'',, P'21:P,,2

where

gij is the genetic covariance

between

xj, and Pij is the phenotypic covariance i Coy(Y3,

Chickens

But if

using YI and Y2 are

AH(Y 1) : i

Efficiency of Selection in Layer-type

x i and

between

xi

and xj, then from equation 4,

H) (8)

ay3

b~Pb 2 hH(Y4)

-

E3, 2 = [(b~Pb3) (D~PB2)]I/2

i Cov(Y4,H) gY4

(9) which reduces to a correlation between

In general,

the efficiency of index YK

relative to an

ues, Y2 and Y3" Hence, E3,2~

index Y1 is,

I and therefore

the index val-

in terms of a correlation, AH(Y2)

~ AH(Y3).

reasoning can be applied to E4, 2" Hence,

AH(Y k )

E4, 2 ~ 1

and ZkH(Y 2) ~ hH(Y4). The efficiency of Y4 relative to Y3 is,

Ek, 1 = Hence,

A similar

the efficiency of Y2 relative to Y1 is

i Coy(H, Y4)aY3 E4, 3 = iCov(H,

lay 2 E2 I- i CYl

Cov (H, Y4)aY3

Y3)ay 4 : Cov(H,Y3)gY4aH

~H =

'

rHy 4 Cunningham

(1969) showed

that

2 : 2 _bl2i/Wii aY2 ~Y1 where Wii is the ith diagonal of the inverse of the P1 matrix corresponding to the ith trait with the index bli that is deleted from the selection index YI" Since b2i/Wii cannot be negative,

the range of values for

2 is 0 ~< 2 ~< 2 Consequently, ~Y2 ~Y2 YI" efficiency of Y3 relative to Y2 is,

E2'I

~< I. The

rHy 3 and is not as determinate as Y4 relative to Y2" However, in choosing between Y3 and Y4' the one whose correlation with H is highest is judged the more efficient index. To summarize,

the relative expected gain from

the four different selection indexes we have defined are in the following order,

i Coy (H, Y3 ) E

AH(Y I) ~ AH(Y 2) b [AH(Y 3) % AH(Y4)] 3, 2

i ay2 ~Y3 If the heritability of the residual is greater than

which in matrix notation is,

b~G3a E3, 2 =

[ (b~P 3b3 ) (b~P 2b2) ] I/2

zero then AH(Y I) >AH(Y2) >AH(Y3) >AH(Y4). If the residual is not genetically correlated with body weight and egg mass, the heritability of feed consumption is increased by ~u -'2"nu' 2. where ~u,2 is the propor-

C.R.

Arboleda,

D.L.

tion of the residual variance

Harris

and A.W.

variance

Nordskog:

Efficiency

in the total phenotypic

of Selection

assumed

in Layer-type

to represent

tion. Finally,

of feed consumption.

From

each

meters The expected evaluated

g~ins from

according

all computations,

the selection

to Equations a selection

20 percent

as parents

of the next generation, chosen

stant in the equations,

of the population was

assumed.

i serves

Be-

only as a con-

the generality

of the results

values paper

genetic

given in Table

(Arboleda

parameters

2. Because

different parameters, recorded

are taken from

values

if any,

indexes.

feed cost from

tions of parameter in Table

3. The

2 and hR

for h 2 ,

a r e t h o u g h t to b e r e a l i s t i c

combination

ty of the residual other selection pected

heritability of

correlations computed

the expected

were

Y1

computed

as given

are presented

Table

increases

increases,

indexes

gain

for the four types of in-

of parameters

index

with

in addition

the different combina-

4 and aid in interpreting

are

we have chosen

estimates

The gain from

assum-

M.

Finally,

b coefficients

and the

have

were

in income

over

u, was

set of the genetic para-

and its genetic

1976)

estimates

and

2, the derived

and egg mass

10 of

different populations

and few,

in the literature,

of arbitrary

et al.,

combination

Table

in the equations

W

to the selection

in Table

The economic

assumed

weight

dex for each

is not altered.

the companion

body

for the popula-

component,

with

given in Table

feed consumption

intensity of i = 1.4,

to selecting

the arbitrarily

were

6, 7, 8 and 9. For

equivalent

cause

indexes

71

the true values

the residual

ed to be uncorrelated Example

Chickens

3. as the heritabili-

but the gains from

are not affected.

result since the genetic variance

the

This is an exof the residual

are

a range Table 2. Arbitrary sen for the example

r G w M . These

although they are not real-

values

of genetic

Parameter

parameters

cho-

Values

ly k n o w n . On t h e o t h e r h a n d , b e c a u s e b o d y w e i g h t i s Heritability : k n o w n to b e h i g h l y h e r i t a b l e , been recorded (h 2 = 0 . 6 0 ) correiations

and many estimates

in the literature,

was used.

Table 3. Expected genetic parameters

of phenotypic

deviations as well as the

0.60 0.05, 0.00,

correlation

0.15 0.20

:

Body weight • mass Body weight X residual Egg mass X residual

were taken from the

-0.20, 0.00 0.00

0.20,

(Arboleda et al. 1976). These are

gain in income

over

feed cost from

Selection h u2

Genetic

coefficients of feed consumption

on body weight and egg mass companion paper

Body weight Egg mass Residual

only a single value

The estimates

and standard

partial regression

have

h2

rGW

0.05

M

different selection

indexes

for different values

of

indexes ~

Y1 ~ Y(IF:W,M,F)

Y2 ~ Y(IF:W,M,r)

Y3 ~ Y(IF:W,M,b)

Y4 Y(I:W,M)

-0.2 0.2 0.6

26.25 12.82 15.91

26.25 12.82 15.91

26.25 12.82 15.91

22.97 5.92 15.39

0.15

-0.2 0.2 0.6

49.84 36.03 44.19

49.84 36.03 44.19

49.84 36.03 44.19

47.97 33.34 43.74

0.05

-0.2 0.2 0.6

26.64 13.60 16.54

26.25 12.82 15.91

26.25 12.82 15.91

22.97 5.92 15.39

0.15

-0.2 0.2 0.6

50.04 36.32 44.42

49.84 36.03 44.19

49.84 36.03 44.19

47.97 33.34 43.74

0.2

See Table 1 for the complete

description

of the selection indexes

0.60

72

C.R.

Arboleda,

D.L. Harris

and A.W.

Nordskog:

Efficiency

of Selection in Layer-type

Chickens

Table 4. Coefficients a for traits in the different selection indexes using different values of h~, h2M and rG ~M

Y(IF:W,M,F) h2u

b3

Y(IF:W,M,r)

Y(IF:W,M,b)

Y(I:W,M)

bl

b2

bl

b2

bl

b2

hM2

rGwM

bl

b2

0.05

-0.2 0.2 0.6

-0.0948 -0.0220 0.0509

0.0033 0.0026 0.0019

0.0000 0.0000 0.0000

-0.0687 -0.0016 0.0655

0.0029 0.0024 0.0018

-0.0687 -0.0016 0.0655

0.0029 0.0024 0.0018

-0.0337 0.0453 0.1243

0.0030 0.0027 0.0024

0.15

-0.2 0.2 0.6

-0.1319 -0.0057 0.1204

0.0089 0.0077 0.0065

0.0000 0.0000 0.0000

-0.1029 0.0134 0.1296

0.0080 0.0071 0.0062

-0.1029 0.0134 0.1296

0.0080 0.0071 0.0062

-0.0741 0.0628 0.1997

0.0090 0.0084 0.0079

0.05

-0.2 0.2 0.6

-0.0830 -0.0101 0.0627

0.0039 0.0032 0.0025

-0.0009 -0.0009 -0.0009

-0.0687 -0.0016 0.0655

0.0029 0.0024 0.0018

-0.0687 -0.0016 0.0655

0.0029 0.0024 0.0018

-0.0337 -0.0453 0.1243

0.0030 0.0027 0.0024

0.15

-0.2 0.2 0.6

-0.1201 0.0061 0.1322

0.0094 0.0083 0.0071

-0.0009 -0.0009 -0.0009

-0.1029 0.0134 0.~296

0.0080 0.0071 0.0062

-0.1029 0.0134 0.1296

0.0080 0.0071 0.0062

-0.0741 0.0628 0.1997

0.0090 0.0084 0.0079

0

0.2

a hi, b2 and b3 are selection index coefficients for body weight, egg mass and feed consumption, respectively. These are the index coefficients used for computing the expected gain from the different selection indexes given in Table 3.

is a component

of the total genetic variance

consumption.

Furthermore,

the residual is not correlated mass,

of feed

Discussion

if the genetic variation of with body weight or egg

it can be effectively utilized only by directly re-

cording feed consumption. When

to 6.0 ~ within the range of parameter

When

from

for improved

0.5%

of genet-

feed efficiency.

h 2 = 0 there is no advantage

to recording

F

U

in preference

to using auxiliary information

in the index (as Y2 and Y3

still show

meters

h M2

and

and h M2

is small

or Y3 ). On the other hand, Y2

a substantial

although this depends

When

and

Y3

over

sitive, the advantage

of the para-

rGw M

is near zero

are much

better than

is much

be related to the change body size for rGW M

less. This seems

near zero.

to more

accurately

the expected

weight and egg mass, correlated

gain

evaluate individual genetThis would include,

of feed correlated

with body

but also a residual component

with differences

and metabolism.

of

it would enable the

in feed efficiency.

not only the components

Even

in efficiency of digestion

if there is no residual genetic

variation in feed consumption,

its measurement

could

be justified on the basis that it will improve

produc-

tion efficiency as an indicator trait (Purser

1960)for

of a particular

output and lower

body weight.

for calculating the importance

variable

an index is to construct

(e.g.,

feed consumption)in

a reduced

index from

the particular variable has been excluded.

to

value of feed consumption

which

Thus, the

can be determined

from

the two indexes,

of its

whereas

should increase

The usual method

Note that in some

to higher feed consumption,

a direct measure

over feed cost because

greater egg mass

in direction of selection for

body size is selected against because

relationship

Y4

rGM w is substantially negative or po-

Y4' but when

indexes,

advantage

on the magnitude

rGw M . Y2

on r G or

considerations

feed consumption in income

ic differences

values chosen.

this might be a useful source

ic variation in breeding

theoretical

breeder

h u2 : 0.2, the gains are small,

Nevertheless,

From

YI = bllXl

+ b12x2

+ b12x3

in and

other indexes,

it is selected for as a genetic indica-

tor for egg mass Although

(Table 4).

Y2 = b21x1

AH(Y 2) ~ AH(Y 3) on theoretical

no difference between

them

could be demonstrated

within the limits of the decimal

+ b22x2

grounds,

places used in Table 3.

When

there is no real residual genetic component,

expected

gain would be the same

for both Y1

the

and Y2"

C.R.

Arboleda,

In this case, tion would

D.L. Harris

and A. W. Nordskog:

the actual measurement

be useful only to estimate

tions between

F and

W

tion from

compared

accord

showed

that the advantage

By

to measuring

a pre-chosen

in Y3 ~Y(IF:W,

rGIG3

with Y4

and

equation,

such as

rG2G3

M, b) = b31x I + b32x 2. In this case,

the breeding

value of F is g3 = ~1gl + [~2g2 and the

net breeding

value is,

sidual were

zero.

X 2 (or W

portance,

and

of Y2

However, over

i.e., alCG1

Y4

assumed

of W

and

Also they showed M

they are in they also

is higher

are large and opposite

i.e., that the correlations

and

73

and their conclusions

with our results.

sign. In our study these were

feed consump-

value of feed consump-

regression

when

Y2

Chickens

in Eaig i. In our notation,

general

M.

traits in the index with

tion is to predict the breeding

trait, G 3, included

F and

value would be increased.

An alternative approach

of Selection in Layer-type

genetic correla-

as an auxiliary trait, the cor-

relation of the income-related the net breeding

of feed consump-

and between

using feed consumption

Efficiency

= a2~G2,

ing G 3 in an index is proportional

to be zero, M

with the re-

that when

in our study)

in

X 1

are of equal im-

the value of includto the magnitude

of a3~G3. H* = a~g I + a~g 2 Acknowledgment

where

a~ : (a I - a3B 1) and a~ : (a 2 - a3~2). Theoretically to or less than relationships

the relative efficiency of Y3 Y2" This is because relationship

the genetic correlations

Practically, to Y

the linear genetic

of g3 to gl and to g2 is approximated

by the linear phenotypic Y2

however,

for Y3

are assumed

the advantage

if the residual

consumption

is small,

but for

to be known.

of Y2

is negligibly small as demonstrated

3 Thus,

is equal

relative in Table 3.

genetic component

of feed

a linear regression

equation

would nearly equal the efficiency in gains made independent

feed consumption

Gjedrem problem aggregate

This study was c a r r i e d out while the senior author was on a doctoral assignment at Iowa State University sponsored by the Rockefeller Foundation and on a field assignment to DeKalb, Illinois, sponsored by the DeKalb AgResearch, Inc., to gather data for the study.

(1972)

presented

He compared traits, X 1 and

records.

considered

the same

here but viewed

genetic value, two indexes,

with

general

in terms

of the

Eaigi, in a selection index. one with two observed

X2, and the other with an unobserved

Received October 20, 1975 Communicated by H. Abplanalp

Literature Arboleda, C.R.; Harris, D.L.~ Nordskog, A.W.: Efficiency of selection in layer-type chickens by using supplementary information on feed consumption. 2. Application to net income, Theoret. and A p p l . G e n e t i c s 4__88, 7 3 - 8 1 ( 1 9 7 6 ) C u n n i n g h a m , E . P . : The r e l a t i v e e f f i c i e n c i e s o f s e l e c t i o n i n d e x e s . A c t a A g r . S c a n d . 19, 4 5 - 4 8 ( 1 9 6 9 ) G j e d r e m , T. : A s t u d y o n t h e d e f i n i t i o n o f t h e a g g r e gate genotype in a selection index. Acta Agric. Scand. 2_22, 1 1 - 1 6 ( 1 9 7 2 ) H a z e l , L . N . : The g e n e t i c b a s i s f o r c o n s t r u c t i n g s e l e c t i o n i n d e x e s . G e n e t i c s 2_8_8, 4 7 6 - 4 9 0 ( 1 9 4 3 ) Purser, A . F . : T h e u s e of c o r r e c t i o n f o r r e g r e s s i o n o n a s e c o n c h a r a c t e r to i n c r e a s e t h e e f f i c i e n c y of s e l e c t i o n . I n : B i o m e t r i c a l G e n e t i c s , K e m p thorne, O. (ed.), 210-214. Oxford: Pergamon P r e s s 1960 Smith, H.F. : A discriminant function for plant select i o n . A n n a l s o f E u g e n i e s 7, 2 4 0 - 2 5 0 ( 1 9 3 9 )

C.R. Arboleda Professor, Animal Science University of Philippines Laguna (Philippines) D.L. Harris formerly geneticst DeKalb AgResearch Inc., DeKalb, Illinois, now with A.R.S.-U.S.D.A. and Purdue University West LaFayette, Indiana 47906 (USA) A.W. Nordskog Dept. of Animal Science Iowa State University of Science Technology Ames,

I o w a 50011 ( U S A )

and